Tilings (Math 285, Winter 2013)

Instructor: Igor Pak, MS 6125, pak@math.

Class Schedule: MWF 12:00-12:50, MS 7608.

Brief outline

We will give an introduction to the subject, covering a large number of classical and a few recent results. The emphasis will be on the main ideas and techniques rather than proving the most recent results in the field. The idea is to to give a guided tour over (large part of) the field to prepare for more advanced results in the future.

Content:

I will be posting most of the literature I will be covering, numbering them roughly in order of the lectures (some lectures are collapsed into one item). A few of these papers will not be taught - they are included the source of additional reading closely related to lecture material.

Domino tilings
- W.P. Thurston, Conway's tiling groups (1990); the original article by Thurston describing his approach.
- J.C. Fournier, Tiling pictures of the plane with dominoes (1997); concise description of the algorithm and complexity applications.
- T. Chaboud, Domino tiling in planar graphs with regular and bipartite dual (1996); a small but insightful well written generalization.
Combinatorial group theory
- J.H. Conway, J.C. Lagarias, Tiling with polyominoes and combinatorial group theory (1990); the original article.
- W.P. Thurston, Groups, Tings and Finite State Automata (1989); unpublished lecture notes describing the connection and giving more context.
Ribbon tilings of Young diagram shapes
- I. Pak, Ribbon tile invariants (2000); the original article.
- S. Fomin, D. Stanton Rim hook lattices (1997); a concise description of the rim hook bijection.
Tile invariants and ribbon tilings of general regions
- C. Moore, I. Pak, Ribbon tile invariants from signed area (2002); the original article using C-L approach.
- R. Muchnik, I. Pak, On tilings by ribbon tetrominoes (1999); here Lemma 2.1 (the "induction lemma") is given with a proof which is omitted in C-L paper.
- M. Korn, Geometric and algebraic properties of polyomino tilings (MIT thesis, 2004); Section 10 gives a clean alternative proof, Section 12 gives a counterexample to the inductive lemma in 3-dim.
- S. Sheffield, Ribbon tilings and multidimensional height functions (2002); proof of 2-connectivity for ribbon tilings and Thurston style tileability algorithm.
Rectangles with one side integral
- S. Wagon, Fourteen Proofs of a Result About Tiling a Rectangle (1988); original article with 14 proofs (not all of them clear or valid)
- D.J.C. MacKay, Simple Proofs of a Rectangle Tiling Theorem, useful variations and additional details on these proofs.
- P. Winkler, Integers and Rectangles, in Mathematical Puzzles (2004), another proof.
- R. Kenyon, A note on tiling with integer-sided rectangles (1994); a new group theory style proof with algorithmic applications.
- N.G. de Bruijn, Filling Boxes with Bricks (1969), the original motivation behind this study.
- V.G. Pokrovskii, Linear relations in dissections into n-dimensional parallelepipeds (1985); an interesting but little cited generalization of de Bruijn's theorem (free access Russian original)
Tilings with two bars
- C. Kenyon, R. Kenyon, Tiling a polygon with rectangles (1992); the original article.
- D. Beauquier, M. Nivat, E. Rémila, M. Robson, Tiling figures of the plane with two bars (1995); a complementary hardness result.
T-tetromino tilings
- D.W. Walkup, Covering a rectangle with T-tetrominoes (1965), tileability of rectangles
- M. Korn, I. Pak, Tilings of rectangles with T-tetrominoes (2004); height function, local move connectivity and counting
Further applications of height functions
- M. Luby, D. Randall, A. Sinclair, Markov chain algorithms for planar lattice structures (2001), height function for 3-colorings of the grid
- R. Kenyon, Tiling a polygon with parallelograms (1992); poly time algorithm.
Tilings of rectangles
- N.G. de Bruijn, D.A. Klarner, A finite basis theorem for packing boxes with bricks (1975), original article (first correct proof in full generality)
- M. Reid, Klarner Systems and Tiling Boxes with Polyominoes (2004), a simplified proof with interesting applications
- J. Yang, Rectangular tileability and complementary tileability are undecidable (2012), a complementary hardness result.
Tilings of rectangles with rectangles
- F.W. Barnes, Algebraic theory of brick packing, part I and part II (1982); original breakthrough article
- M. Reid, Asymptotically Optimal Box Packing Theorems (2008), a Klarner system style proof and some amazing examples
- T. Lam, Tiling with commutative rings (2008), an intro to commutative ring method
- O. Bodini, Tiling a Rectangle with Polyominoes (2003), an algebraic proof, barely readable.
Order of tiles
- D.A. Klarner, Packing a rectangle with congruent N-ominoes (1969); a number of interesting examples.
- M. Reid, Tiling rectangles and half strips with congruent polyominoes (1997); more examples and discussion of the odd order conjecture.
Augmentability
- Yang's paper (see above); hardness of augmentability and decidability of augmentability with rectangles.
- Korn's thesis (see above), Section 11; negative solution of augmentability for dominoes.
Valuations
- C. Freiling, D. Rinne, Tiling a square with similar rectangles (1994); necessary and sufficient conditions for tiling squares with rectangles similar to a given.
- M. Laczkovich, G. Szekeres, Tiling of the square with similar rectangles (1995); an independent proof.
- C. Freiling, M. Laczkovich, D. Rinne, Rectangling a rectangle (1997); a generalization to rectangular regions.
- M. Prasolov, M. Skopenkov, Tiling by rectangles and alternating current (2010); an alternative proof.
Counting domino tilings and perfect matchings
- L. Lovász, M.D. Plummer, Matching Theory, Chapter 8; among other things, general Pfaffian-Determinant approach.
- R. Kenyon, An introduction to the dimer model (2002); variations on Kasteleyn method, isoradial graphs.
- R.W. Kenyon, J.G. Propp, D.B. Wilson, Trees and Matchings (2000); Temperley's bijection and applications.
Aztec diamond
- A. Björner, R.P. Stanley, A combinatorial miscellany (2010); domino shuffling.
- N. Elkies, G. Kuperberg, M. Larsen, J. Propp, Alternating-sign matrices and domino tilings; the original article with four proofs.
- Jim Propp, San Diego talk slides, outline of a short proof based on the "urban renewal".
- S.-P. Eu and T.-S. Fu, A simple proof of the Aztec diamond theorem (2005), an elegant short proof based on the non-intersecting paths principle.
- K.P. Kokhas, Domino tilings of aztec diamonds and squares (2008); yet another beautiful proof of the Aztec diamond formula via a general recurrence relation (§ 2.1 and 2.2)
- L. Pachter, Combinatorial Approaches and Conjectures for 2-Divisibility Problems Concerning Domino Tilings of Polyominoes (1997); simple proof of divisibility of the number of domino tilings of a even square.

Warnings: Most of these links are external, some are by subscription, some can be broken; occasionally, their content is unverified. Also, the explanations are NOT review, but rather quick summary of material I used from the sources; often there is wealth of other work presented there as well.

General references:

Federico Ardila and Richard P. Stanley, Tilings (2005), an introductory article.
Branko Grunbaum and G. C. Shephard, Tilings and Patterns (1986), an instant classic. A generic introduction to the subject.
Solomon W. Golomb, Polyominoes: Puzzles, Patterns, Problems, and Packings, true classic in the recreational literature (Second Ed., 1995).

Click here to return to Igor Pak Home Page.

Last updated 3/8/2013